In previous weeks, demonstrations have begun only after sundown and the number of people at this Friday's protests appeared smaller than in the past.
On Sunday, protest organisers called for demonstrations in Khartoum and other towns, including Madani, Kosti and Dongola, as part of what they have called a "Week of Uprising".
Qatar's ruler, Sheikh Tamim bin Hamad Al Thani, called Al Bashir shortly after the protests began to offer his support.
Reuters witnesses said security forces used tear gas against dozens of demonstrators in Al-Halfaya Bahri in southern Khartoum and against a separate demonstration by dozens of people who had emerged from Sayyed Abdel-Rahman Mosque in Omdurman, on the other side of River Nile to the capital. The rest were killed in Khartoum's twin city of Omdurman and regions north and northeast of the capital.
Authorities say at least 24 people have died in the protests, while rights group Human Rights Watch has put the death toll at 40, including children and medical staff.
"Patients and doctors in Omdurman Hospital were attacked with tear gas and live bullets as Sudanese security forces ramped up their mission to suppress peaceful protests", Sarah Jackson, Amnesty's deputy director for East Africa, the Horn and the Great Lakes, said on Thursday.
The protests have since swiftly escalated into nationwide rallies widely seen as the biggest threat to Bashir's rule in his three decades in power.
Repeated shortages of food and fuel have been reported in several cities, including Khartoum, while the costs of foods and medicines have more than doubled.
The crackdown has drawn worldwide criticism, with countries like Britain, Norway, Canada and the United States warning Khartoum that its actions could "have an impact" on its relations with their governments.
Last week, Sudanese security sources confirmed the presence of Wagner Russian military in the country and that they work with the National Intelligence and Security Services (NISS) but refused to speak about their activities in Sudan.
The United States lifted 20-year-old trade sanctions on Sudan in October 2017, but many investors continue to shun a country still listed by Washington as a state sponsor of terrorism.
The foreign currency shortages began with the 2011 secession of South Sudan, which took with it the bulk of oil revenues.